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Jeffery–Hamel flow

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In fluid dynamics Jeffery–Hamel flow is a flow created by a converging or diverging channel with a source or sink of fluid volume at the point of intersection of the two plane walls. It is named after George Barker Jeffery(1915)[1] and Georg Hamel(1917),[2] but it has subsequently been studied by many major scientists such as von Kármán and Levi-Civita,[3] Walter Tollmien,[4] F. Noether,[5] W.R. Dean,[6] Rosenhead,[7] Landau,[8] G.K. Batchelor[9] etc. A complete set of solutions was described by Edward Fraenkel in 1962.[10]

Flow description

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Consider two stationary plane walls with a constant volume flow rate is injected/sucked at the point of intersection of plane walls and let the angle subtended by two walls be . Take the cylindrical coordinate system with representing point of intersection and the centerline and are the corresponding velocity components. The resulting flow is two-dimensional if the plates are infinitely long in the axial direction, or the plates are longer but finite, if one were neglect edge effects and for the same reason the flow can be assumed to be entirely radial i.e., .

Then the continuity equation and the incompressible Navier–Stokes equations reduce to

The boundary conditions are no-slip condition at both walls and the third condition is derived from the fact that the volume flux injected/sucked at the point of intersection is constant across a surface at any radius.

Formulation

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The first equation tells that is just function of , the function is defined as

Different authors defines the function differently, for example, Landau[8] defines the function with a factor . But following Whitham,[11] Rosenhead[12] the momentum equation becomes

Now letting

the and momentum equations reduce to

and substituting this into the previous equation(to eliminate pressure) results in

Multiplying by and integrating once,

where are constants to be determined from the boundary conditions. The above equation can be re-written conveniently with three other constants as roots of a cubic polynomial, with only two constants being arbitrary, the third constant is always obtained from other two because sum of the roots is .

The boundary conditions reduce to

where is the corresponding Reynolds number. The solution can be expressed in terms of elliptic functions. For convergent flow , the solution exists for all , but for the divergent flow , the solution exists only for a particular range of .

Dynamical interpretation

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Source:[13]

The equation takes the same form as an undamped nonlinear oscillator(with cubic potential) one can pretend that is time, is displacement and is velocity of a particle with unit mass, then the equation represents the energy equation(, where and ) with zero total energy, then it is easy to see that the potential energy is

where in motion. Since the particle starts at for and ends at for , there are two cases to be considered.

  • First case are complex conjugates and . The particle starts at with finite positive velocity and attains where its velocity is and acceleration is and returns to at final time. The particle motion represents pure outflow motion because and also it is symmetric about .
  • Second case , all constants are real. The motion from to to represents a pure symmetric outflow as in the previous case. And the motion to to with for all time() represents a pure symmetric inflow. But also, the particle may oscillate between , representing both inflow and outflow regions and the flow is no longer need to symmetric about .

The rich structure of this dynamical interpretation can be found in Rosenhead(1940).[7]

Pure outflow

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For pure outflow, since at , integration of governing equation gives

and the boundary conditions becomes

The equations can be simplified by standard transformations given for example in Jeffreys.[14]

  • First case are complex conjugates and leads to

where are Jacobi elliptic functions.

  • Second case leads to

Limiting form

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The limiting condition is obtained by noting that pure outflow is impossible when , which implies from the governing equation. Thus beyond this critical conditions, no solution exists. The critical angle is given by

where

where is the complete elliptic integral of the first kind. For large values of , the critical angle becomes .

The corresponding critical Reynolds number or volume flux is given by

where is the complete elliptic integral of the second kind. For large values of , the critical Reynolds number or volume flux becomes .

Pure inflow

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For pure inflow, the implicit solution is given by

and the boundary conditions becomes

Pure inflow is possible only when all constants are real and the solution is given by

where is the complete elliptic integral of the first kind.

Limiting form

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As Reynolds number increases ( becomes larger), the flow tends to become uniform(thus approaching potential flow solution), except for boundary layers near the walls. Since is large and is given, it is clear from the solution that must be large, therefore . But when , , the solution becomes

It is clear that everywhere except in the boundary layer of thickness . The volume flux is so that and the boundary layers have classical thickness .

References

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  1. ^ Jeffery, G. B. "L. The two-dimensional steady motion of a viscous fluid." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 29.172 (1915): 455–465.
  2. ^ Hamel, Georg. "Spiralförmige Bewegungen zäher Flüssigkeiten." Jahresbericht der Deutschen Mathematiker-Vereinigung 25 (1917): 34–60.
  3. ^ von Kármán, and Levi-Civita. "Vorträge aus dem Gebiete der Hydro-und Aerodynamik." (1922)
  4. ^ Walter Tollmien "Handbuch der Experimentalphysik, Vol. 4." (1931): 257.
  5. ^ Fritz Noether "Handbuch der physikalischen und technischen Mechanik, Vol. 5." Leipzig, JA Barch (1931): 733.
  6. ^ Dean, W. R. "LXXII. Note on the divergent flow of fluid." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 18.121 (1934): 759–777.
  7. ^ a b Louis Rosenhead "The steady two-dimensional radial flow of viscous fluid between two inclined plane walls." Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. Vol. 175. No. 963. The Royal Society, 1940.
  8. ^ a b Lev Landau, and E. M. Lifshitz. "Fluid Mechanics Pergamon." New York 61 (1959).
  9. ^ G.K. Batchelor. An introduction to fluid dynamics. Cambridge university press, 2000.
  10. ^ Fraenkel, L. E. (1962). Laminar flow in symmetrical channels with slightly curved walls, I. On the Jeffery-Hamel solutions for flow between plane walls. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 267(1328), 119-138.
  11. ^ Whitham, G. B. "Chapter III in Laminar Boundary Layers." (1963): 122.
  12. ^ Rosenhead, Louis, ed. Laminar boundary layers. Clarendon Press, 1963.
  13. ^ Drazin, Philip G., and Norman Riley. The Navier–Stokes equations: a classification of flows and exact solutions. No. 334. Cambridge University Press, 2006.
  14. ^ Jeffreys, Harold, Bertha Swirles, and Philip M. Morse. "Methods of mathematical physics." (1956): 32–34.